What is the integral of x/(5+2x) ?

The integral of x/(5+2x) is 1/4 (5+2x-5ln|5+2x|)+C

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integral of x/(5+2x)

\int \frac{x}{5 + 2 x} d x

\frac{1}{4} (5 + 2 x - 5 ln ∣ 5 + 2 x ∣) + C

Solution steps

\int \frac{x}{5 + 2 x} d x

Apply u-substitution

= \int \frac{u - 5}{4 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{4} \cdot \int \frac{u - 5}{u} d u

Expand

\frac{u - 5}{u} : 1 - \frac{5}{u}

= \frac{1}{4} \cdot \int 1 - \frac{5}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{4} (\int 1 d u - \int \frac{5}{u} d u)

\int 1 d u = u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= \frac{1}{4} (u - 5 ln ∣ u ∣)

Substitute back

u = 5 + 2 x

= \frac{1}{4} (5 + 2 x - 5 ln ∣ 5 + 2 x ∣)

Add a constant to the solution

= \frac{1}{4} (5 + 2 x - 5 ln ∣ 5 + 2 x ∣) + C

What is the integral of x/(5+2x) ?
The integral of x/(5+2x) is 1/4 (5+2x-5ln|5+2x|)+C